'Best to get it out and be done with it," Joe Goddard tweeted to me the other day. "Otherwise there's just a geeky elephant in the room throughout." The geeky elephant is the certainty that in any interview with Hot Chip, the writer – adopting the high-handed tone of some Victorian missionary inquiring as to whether this latest heathen tribe really does practise cannibalism – will ask the band if they are indeed geeks, or perhaps, at a stretch, nerds. At which point Hot Chip will roll their eyes, sigh, and explain that, no, they are not geeks, they are just not awfully fussed about stylists.

The thing is, when I try to sidle my way into this topic, halfway through our interview, I don't want to accuse them of geekery. In fact, I struggle desperately to avoid using the word "geek". To Goddard and Hot Chip's co-leader Alexis Taylor, though, it must appear as though I'm just racking my brain for a synonym. The two men – friends since school – stare at me in silence, waiting for the word to spill out.

What I want to say is that Hot Chip don't look like geeks to me. At this point, five albums into their career, they look – and sound – increasingly like one of the great British pop groups. They look like a gang – not a street gang, not a tough gang, but the slightly baffling boys who'd be in the corner of the school dining room, and who turn out years later to have been cooler than the rest of the sixth form put together.

It helps that they all look different – who said gangs needed to be identical? – as if each of them expresses some part of the collective Hot Chip personality. Of the two frontmen – who share a sometimes alarming taste in garish sportswear – Goddard is big and garrulous, Taylor small and self-contained. Their three bandmates – generally besuited – seem to occupy different roles. Felix Martin looks like an eccentric inventor, Al Doyle like a bedheaded partier, Owen Clarke like the token matinee idol. Watching them have their photograph taken for the Observer – a process borne with varying degrees of amusement by the band members – it's hard not to think: Hot Chip look peculiar, but they look great.

"I think you're one of the only people who thinks that," Taylor says when we sit down afterwards. Goddard just looks astonished before explaining that the "geek" thing is one of many misreadings of Hot Chip made in 2004 when they released their first album, a tribute to the US R&B and hip-hop they loved.

"If a group of people are going to bring out a record like Coming on Strong, maybe they'd really consciously try to put together an image that suggested they were into US hip-hop," he says. "But we were middle-class white kids from Putney and we never tried to hide that. We just had a love of Destiny's Child and the Beastie Boys. People are used to bands that tell you what to think about them by all dressing in leather jackets and Converse. And we just never did it."

If they weren't geeks, they faced the great catch-all insult of the last decade: they were hipsters. A reviewer of a 2004 live show wrote: "A little bird tells me that there's a degree of Hoxtonite irony at play here, and that Hot Chip actually spend their spare time listening to such hyphenated genres as lo-fi, alt-country and post-rock." As if Taylor couldn't possibly have been just expressing the depths of his own love of Prince when he sang: "I'm sick of motherfuckers trying to tell me that they're down with Prince." As if the spindly sound of their debut was some conceptual joke at the expense of American R&B, rather than a reflection of the fact that it's hard to make a Timbaland record on a laptop in a suburban bedroom.

What actually marks out Hot Chip, and something they have mastered on album five, In Our Heads, is an empathetic warm-heartedness. They are sincere: for all the detachment of Taylor's falsetto, the cold squiggles of the synths, the sense that clever people are at play, Hot Chip actually remake soul music for the English middle class, swapping unbridled passion for diffident understatement.

"It was funny to be so misunderstood at the beginning," Taylor says, "because that wasn't something people picked up on. I thought it was obvious the music was open-hearted."

"There's more humour and silliness on the first record," Goddard says, trying to explain why it was misread.

The vast distance between Hot Chip and those who use music as an ironic prop is made evident when they talk about current pop, and the motivations of its makers. "There's quite a lot of cynicism now about how to make pop records and what the point of it is," Taylor says. "I saw the lady from N-Dubz on a chatshow and they were asking how she felt about the band splitting up. She just talked about having to pay her mortgage being the main issue."

Goddard complains about the lack of imperfection in modern pop, about it being full of records that "feel like they've come from a factory that tries to correct everything and takes out all the flaws that make everything really lovable for me. Pop music's become quite conservative in a lot of ways."

Hang on. Isn't that reading ignoring the reality of pop's past? Think back to the late 80s, to Stock, Aitken and Waterman's Hit Factory churning out identikit singles by singers whose sole function was to look nice for the sleeve. Think back further to the svengalis of the 50s, 60s and 70s, whose sole concern was maximising production to maximise profit.

"Yeah," Goddard admits. "Yeah, that's true."

"There are good examples of hit factories working, though – Motown and Phil Spector," Taylor says. "I'm talking about the motivation of the people who are actually the pop stars, loving records and making records because of that. I think it was more to do with whether people have a fascination with the sounds of records."

"You can make good records on computers," Goddard concludes, "but it's down to your intention."

Listening back to our conversation, it's remarkable how much of it – whatever the ostensible tack of any particular question – ends up being about the state of pop music. Taylor and Goddard seem to be always comparing Hot Chip to prevailing trends, and finding the prevailing trends wanting. They formed, Taylor says, because "we were quite dismissive of other music around"; they put a lot of work into being a good live band, Goddard says, "and it seems increasingly anachronistic, because most electronic pop acts would be a laptop and a singer".

They look at the recent rash of nostalgia for first-wave rave culture and worry that dance music, like rock music, has just become another recyclable nostalgia commodity.

"I think it's difficult for a musical movement happening now to be such a widespread thing, where everything else is swept aside," Goddard says with an air of regret. "When that happened at the beginning of rave, it was the most exciting thing in popular culture and it swept everyone along. Maybe that's what makes it exciting to the people that are reviving it, the fact that it cleared the decks. Like the Chase & Status video [for the 2011 single Blind Faith] – everyone in the pub is part of this thing that's going to happen; they're all going off to the rave.

"Now, if you're in the pub with your mates, there'll be people going to a jungle night, people going to a garage night. There's not one thing that everyone's into." He looks up, like a big, sad St Bernard. "But you can't stop the march of progress."

The curious thing, though, is that Hot Chip are, as they recognise, the embodiment of that move away from the single hegemonic pop trend. Their own music contains many of the varied strands of dance music, steals from hip-hop and R&B, with melody lines taken from English folk-rock, attitudes from post-punk. Don't Deny Your Heart, one of the stand-outs from In Our Heads, could be a great lost 80s mainstream pop hit, with its bold sonic building blocks held together by the mortar of Al Doyle's Chic-aping guitar.

Then there are the external projects that feed back in, notably Goddard's 2 Bears house revivalism and Taylor's adventures in improvisation with About Group. Somehow, through it all, they have contrived a signature sound. No one else sounds quite like Hot Chip.

"I think when we started out we were quite conscious of the idea that it had to sound like it was new in some way," Taylor says. "Obviously, I can now see there are things other people hear in our music and it's not as original as we might want it to be. But what's very important to how we make music is layering things together so it doesn't just sound like the most obvious things you could put on a record."

And when something big and obvious is taken, its chemistry is altered so it ceases to be a steal and becomes an invention – like the Detroit techno riff at the end of 2010's I Feel Better, played on steel pans, with the result that it sounds like a cyberman walking through carnival.

With In Our Heads Hot Chip return to the indie sector after three albums with EMI – it's coming out on Domino, for which Taylor used to work way back when. They were there as the venerable label unravelled before finally being sold off, and it wasn't a happy experience. "The last time we put a record out it didn't really seem like EMI knew what it was doing any more, especially with our band," Taylor says. "EMI did good things with our band, and I don't want to make out it's a major v indie thing, but it definitely was a bad time to still be on that label."

"I don't think there was anyone left who was there at the start," Goddard says. "Our A&R man decided to become a chef and went on MasterChef. He got pretty far. People really drifted away and they were often the people who were most passionate about good records. EMI treated us as a priority for a while, and we did OK, but we never sold as many records as they would have liked us to. And then they realised that and stopped treating us as a priority."

"They have a history," Taylor adds. "Kraftwerk and Kate Bush and the Beatles. But I started to notice that all the bands I really liked on EMI were from a long time ago."

Even if EMI were never quite happy with Hot Chip's level of success, they must have noticed the group's inexorable rise to national treasure status – the Grammy and Mercury nominations, the top 10 albums, the sense that a new Hot Chip album was an event. What disappointed EMI has been incalculable success to Taylor.

"If a record was released by a company, for me that was what I was trying to have happen. I didn't have grander ambitions than

being able to record music that got released. This small label called Victory Garden put out our first EP [in 2000] and that was enough to make me feel that what we were doing was being recognised by someone."

Finally, though, does EMI's share price, indie v major cred wars or the state of the world really have very much to do with Hot Chip – with this world of their own making that Taylor and Goddard have taken from their classrooms and bedrooms to the centre of British pop?

"It's a weird dream world when you're making music," Taylor says. "You're just exploring things that are of interest to you. Why would that be something to take seriously?"

Hot Chip headline Camp Bestival in July. Hear In Our Heads a week before release at guardian.co.uk/music from June 4